Room-temperature sub-100 nm Néel-type skyrmions in non-stoichiometric van der Waals ferromagnet Fe3-xGaTe2 with ultrafast laser writability

Realizing room-temperature magnetic skyrmions in two-dimensional van der Waals ferromagnets offers unparalleled prospects for future spintronic applications. However, due to the intrinsic spin fluctuations that suppress atomic long-range magnetic order and the inherent inversion crystal symmetry that excludes the presence of the Dzyaloshinskii-Moriya interaction, achieving room-temperature skyrmions in 2D magnets remains a formidable challenge. In this study, we target room-temperature 2D magnet Fe3GaTe2 and unveil that the introduction of iron-deficient into this compound enables spatial inversion symmetry breaking, thus inducing a significant Dzyaloshinskii-Moriya interaction that brings about room-temperature Néel-type skyrmions with unprecedentedly small size. To further enhance the practical applications of this finding, we employ a homemade in-situ optical Lorentz transmission electron microscopy to demonstrate ultrafast writing of skyrmions in Fe3-xGaTe2 using a single femtosecond laser pulse. Our results manifest the Fe3-xGaTe2 as a promising building block for realizing skyrmion-based magneto-optical functionalities.

This paper is well written and presented a new phase, Fe-deficiency Fe3-xGaTe2, observing Néel-type skyrmions.I recommend for publication with minor revision.
-In terms of Fe composition, EDS is never a good quantitative way to determine the composition.The accuracy will be improved if the composition of perfect Fe3GaTe2 crystal flake is used as reference to compare with the Fe-deficient flake using the same experimental conditions.
-The space group changed to non-inversion symmetry based on the XRD and HAADF-STEM image.It will be more convincing if additional data using CBED to confirm the space group.
-The Fe columns in Fig. 1f are not really clear.The atomic columns do not look clean.A better clear HAADF-STEM image will be better that shows better Fe Columns.Would the author claim that the Fe deficiency occurs only at those FeII positions?-Can the author control the Fe content?
Reviewer #2 (Remarks to the Author): Reviewer comments on "Room-temperature sub-100 nm Néel-type skyrmions in nonstoichiometric van der Waals ferromagnet Fe3-xGaTe2 with ultrafast laser writability" In this work, the authors demonstrate the presence of Néel skyrmions in FGT due to a DMI tied to broken inversion symmetry from Fe vacancies.Additionally, the authors demonstrate the ability to drive the phase change optically, by locally heating the sample in-situ with a 520nm pulsed laser during LTEM measurement.
While the material parameters are marginally higher than previously reported in 2D vdW skyrmion systems, it seems to me that the novelty of this work comes from the mechanism/crystallography, which should be better described.The physics of the material is the new discovery here, as the material properties are almost the same as existing systems.I think this work needs to be more motivated by, and discuss, the physics of the system, rather than just experimental observations.In, for example, (Fe0.5Co0.5)3GeTe2: • The presence of Neel skyrmions and THE of approximately the same value has previously been reported (Zhang et al.Room-temperature skyrmion lattice in a layered magnet (Fe0.5Co0.5)5GeTe2.Sci. Adv.8,eabm7103(2022).) • The ordering of the skyrmion lattice has previously been reported (Meisenheimer et al.Ordering of room-temperature magnetic skyrmions in a polar van der Waals magnet.Nat Commun 14, 3744 (2023).) • A similar phase diagram and thickness dependence has been reported (Zhang et al.Room-temperature skyrmion lattice in a layered magnet (Fe0.5Co0.5)5GeTe2.Sci. Adv.8,eabm7103(2022).) • Approximately the same Curie temperature and similar mechanism (ordering of empty Fe sites) has been reported (Zhang et al.A room temperature polar magnetic metal.Phys.Rev. Materials 6, 044403 (2022).)And while the in-situ measurement is new, its value would come from actual dynamical measurements-by doing just quasistatic quenching, it doesn't seem like there is functionally any difference between just T,B cycling without the optical pump (Meisenheimer et al.Ordering of room-temperature magnetic skyrmions in a polar van der Waals magnet.Nat Commun 14, 3744 (2023), Zhang et al.Room-temperature skyrmion lattice in a layered magnet (Fe0.5Co0.5)5GeTe2.Sci. Adv.8,eabm7103(2022)., Zhang et al.Room-temperature skyrmion lattice in a layered magnet (Fe0.5Co0.5)5GeTe2.Sci. Adv.8,eabm7103(2022).)Especially so because you motivate the experiment from the perspective of "ultrafast writing of skyrmions" (L 104, L 313).
More specifically, there needs to be more discussion on the mechanism of the DMI.A global parameter implies that the empty Fe sites are ordering?DFT is used to simulate the value of DMI, but is the relaxed structure comparable?Does this value match with what is measured (does it give skyrmions/domains of similar size)?What does the anisotropy of the parent compound look like?
Having an in-plane P, to my knowledge, separates this from the existing work, but the parameters are ultimately largely the same?How does the directionality of B change the shape of the skyrmions?It seems like the interaction with D should be unique.
Why is there such a large variance in the sizes of the skyrmions?this is also different to what is generally reported.In fact, it almost looks bimodal in many images.
There needs to be more interpretation of the results and tying back to a structure-property relation for me to be comfortable recommending this paper.
Smaller notes: You mix cubic and hexagonal coordinates-since the system is hexagonal, you should make this consistent (e.g.[0001] instead of [001], L 138) You need to soften some statements in the introduction-e.g.rapid thermal annealing is not going to "revolutionize skyrmion logic" (L103), these processes have been around for a while and, additionally, are not particularly chip compatible.

Reviewer #3 (Remarks to the Author):
General Comment: The authors investigate a non-stoichiometric room temperature magnet Fe2.86GaTe2 crystal where the Fe vacancies induce the formation of DMI by spatial inversion symmetry breaking.Such an in-plane isotropic DMI brings about RT Néel-type skyrmions, and the size of the skyrmions can be regulated by the sample thickness and the external magnetic field.The dynamic writing process of RT skyrmions in Fe2.86GaTe2 flakes enhances the potential application of spintronic devices.The paper is timely and of interest.

Referee A's Comment 1:
In terms of Fe composition, EDS is never a good quantitative way to determine the composition.The accuracy will be improved if the composition of perfect Fe3GaTe2 crystal flake is used as reference to compare with the Fe-deficient flake using the same experimental conditions.
Author's reply: We highly appreciate the referee's suggestions to enhance the accuracy of composition determination.In order to obtain fully stoichiometric Fe3GaTe2, we have systematically grown a series of Fe3-xGaTe2 single crystals by varying the Fe content in the raw material composition, utilizing a Te-flux method.Subsequently, comprehensive energy-dispersive X-ray spectroscopy (EDS) mapping was conducted on the cleaved surfaces of these crystals to determine their chemical composition.To ensure the reliability of the EDS results, mapping was carried out at four distinct areas for each sample.Utilizing the Ga ratio as the normalization factor and ignoring the variation of Te content, their corresponding chemical composition were established.Table R1 provides a comprehensive overview of the raw material composition and the final crystal composition.It is clearly observed that when the raw Fe ratios fall below 0.8, the Fe3-xGaTe2 phase cannot be formed.Instead, a mixture of phases, including GaTe and Ga2Te3, is produced.When the raw Fe ratios are equal to or greater than 0.9, Fe3-xGaTe2 single crystals are crystallized, with the Fe content in these single crystals increasing proportionally with the raw Fe ratios.However, an increase in the raw Fe ratios to 1.3 results in the formation of FeTe phases.As surmised in Table R1, we find that the chemical formulas for the crystals with the minimum and maximum Fe content correspond to Fe2.84±0.05GaTe2and Fe2.96±0.02GaTe2,respectively.This result indicates that Fe vacancies always exist in the single crystals synthesized using a Te-flux method.
The formula of the later one, i.e.Fe2.96±0.02GaTe2, is quite close to that of the perfect stoichiometric Fe3GaTe2 crystal, and can be regarded as the reference to improve the accuracy suggested by the referee.In the previous version of our manuscript, we reported the observation of Néel-type skyrmions in the Fe3-xGaTe2 single crystals synthesized with a raw Fe ratio of 0.9.To highlight the existence of Fe vacancies, the chemical formula was denoted as Fe2.86GaTe2, which corresponds to the minimum Fe content determined by the EDS mapping.In the revised manuscript, to enhance the accuracy of the chemical formula, an error bar has been added by summarizing the EDX results obtained at different areas, and the chemical formula is denoted as Fe2.84±0.05GaTe2(see Page 3, Lines 118-121 in the main text).Simultaneously, in accordance with the referee's suggestion, the EDX results of the crystals with other Fe content and associated analysis are also presented in Supplementary Note 1 and Table S1 for reference.
Table R1.Summary of the raw material composition and the final product for the growth of Fe3-xGaTe2 samples using the self-flux method.Author's reply: We agree with the referee that Convergent Beam Electron Diffraction (CBED) is an efficient approach to characterize the non-inversion symmetry of the crystal structure.Following the referee's suggestion, we have performed CBED measurements with the electron beam injected along the [112 ̅ 0] zone axis, which is the only zone axis that allows for directly observing significant displacement of Feii columns.However, the obtained diffraction disks are seriously overlapped (as shown in Fig. R1a), despite various optimizations, including tuning the camera length, electron beam size, and beam-convergence angle α.Therefore, instead of utilizing CBED, we have performed selected area electron diffraction (SAED) measurements along both the [112 ̅ 0] and [101 ̅ 0] zone axes to confirm the non-inversion structural symmetry, as shown in Figs.R2a and R2b.It is clearly observed that the SAED patterns exhibit a series of (000l) diffraction patterns, such as (0005 ̅ ) , (0001 ̅ ) , ( 0003) and (0007) .
However, the simulations demonstrate that the odd l (l = 2n + 1) values of (000)   diffractions are allowed for the non-centrosymmetric space group P3m1, but forbidden in a centrosymmetric space group P63/mmc.To further confirm the differences, we simulated SAED patterns based on the XRD refined non-centrosymmetric structure of Fe3-xGaTe2 (space group P3m1, with Feii deviation), which align exceptionally well with the experimental results (Fig. R2c and R2d).Instead, a prefect centrosymmetric structure of Fe3GaTe2 (space group P63/mmc, without Feii deviation), lacks (000l) and (hh2h ̅̅̅ l) diffraction patterns for the odd values of l (Fig. R2c and R2d).R4a . For a quantitative determination of the deviation of the Feii atoms, we focused on the region marked by the blue rectangles (left panel of Fig. R4a) comprising Te-Feii-Te atoms.We then vertically integrated the corresponding imaging intensity line profile (Fig. R4b).By referencing the center of the two Te atoms, the deviation of the Feii atom towards the c direction was determined to be 0.20 Å. Utilizing the same procedure, we surveyed an area of 2 × 17 unit cells, yielding an average Feii deviation of 0.16  0.06 Å.
Additionally, we observed that the image intensity of Fei-a above Feii is weaker than that of Fei-b below Feii, as evident in the imaging intensity line profile of Fei-a-Fei-b atoms in Fig. R4c.Since imaging intensity is generally proportional to the number of projected atoms [PNAS 107.26 (2010): 11682-11685], the contrast difference between Fei-a and Fei-b indicates asymmetric site occupations, suggesting a small quantity of Fe vacancies in the Fei-a site.In the previous version of our manuscript, we emphasized the Fe deficiency at Feii sites with an occupancy ratio of 0.8467.Upon re-evaluating the results of the refined single-crystal XRD, we discovered that the lower Fei-b sites are nearly fully occupied, while the higher Fei-a sites have an occupancy ratio of 0.9688.
Thus, the Fe deficiency occurs not only at the Feii positions, but also at some Fei-a positions.
In the revised manuscript, we have added the associated discussions on Feii deviations, Feii vacancies and the asymmetric Fei vacancies (refer to Page 7, Lines 187-198 and Page 6, Lines 154-159 in the main text) into the main text and Supplementary Figs. S4 and S5, as also shown below.
"However, the HAADF image along the [112 ̅ 0] zone axis (Fig. 1f) reveals that Feii atoms deviate clearly from the center position of the Te slices along the c-axis, which is also supported by the annular bright-field (ABF-STEM) image in Fig. S4.By referencing the center of the two Te atoms in a magnified ABF-STEM image (Fig. S5), an averaged Feii deviation is calculated as 0.16 ± 0.06 Å over an area of 2 × 17 unit cells (see Supplementary Note 2)." "In comparison to the stoichiometric Fe3GaTe2 with a centrosymmetric structure, the presence of Fe deficiency in Fe2.84±0.05GaTe2should exert a pivotal influence on the Feii deviation for the asymmetric structure.Our refined single-crystal XRD indicates that Fe deficiency is predominantly concentrated at the Feii positions with an occupancy ratio of 0.8467.Additionally, the upper-layer Fei-a sites have an occupancy ratio of 0.9688, while the under-layer Fei-b sites are nearly fully occupied (Fig. 1h).As observed in the line profile of Fei-a and Fei-b atoms in the ABF-STEM image (Fig. S5c), it is apparent that the image intensity of Fei-a above Feii is weaker than that of Fei-b below Feii.Since the ABF imaging intensity is generally proportional to the number of projected atoms 41 , the contrast difference between Fei-a and Fei-b indicates asymmetric site occupations, suggesting a small quantity of Fe vacancies in the Fei-a site, which is consistent with the results of single-crystal XRD."  Author's reply: We sincerely appreciate the insightful comments provided by the referee.Following the comments, we have further systematically grown a series of Fe3-xGaTe2 single crystals by varying the Fe content in the raw material composition, utilizing a Te-flux method.A comprehensive overview of the raw material composition and the final product is outlined in Table R1 (see Author's reply to Referee A' Comment 1).It is clearly demonstrated that controlling the Fe content in the final crystals is achievable by varying the raw Fe ratio.Specifically, when the raw Fe ratio falls below 0.8, the Fe3-xGaTe2 phase cannot be formed.Instead, a mixture of phases, including GaTe and Ga2Te3 phases, is produced.In contrast, when the raw Fe ratio is equal to or greater than 0.9, Fe3-xGaTe2 single crystals can be crystallized, with the Fe content in these single crystals increasing proportionally with the raw Fe ratio.However, an increase in the raw Fe ratio to 1.3 results in the formation of FeTe phase.In the revised manuscript, we have incorporated discussions on controlling the Fe content of the crystals into both the main text (refer to Page 5, Lines 110-120) and Supplementary information (refer to Supplementary Note 1 and Table S1), providing a more comprehensive understanding of the growth conditions and their impact on the final composition of the single crystals.
"In order to control the Fe content, we systematically grew a series of Fe3-xGaTe2 single crystals by varying the Fe content in the raw material composition, utilizing a Te-flux method (see Methods section and Supplementary Note 1).To determine the chemical composition of the as-grown crystals, energy dispersive X-ray spectroscopy (EDX) analyses were conducted on the surfaces of Fe3-xGaTe2 nanoflakes (Fig. 1a) that were exfoliated and placed onto the Si3N4 membrane (see Methods).The ratio of raw materials and the corresponding final crystal composition are listed in Table S1, Supplementary Fig. S1 and Fig. 1b.We found that the Fe deficiencies always exist in these crystals, while the minimum and maximum Fe contents correspond to Fe2.84±0.05GaTe2and Fe2.96±0.02GaTe2,respectively.This result implies the feasibility of inducing Fe deficiency in the samples.To highlight the existence of Fe vacancies, the subsequent studies were focused on the minimum Fe content sample Fe2.84±0.05GaTe2." Response to the Report of Referee B Referee B's General Comment: In this work, the authors demonstrate the presence of Néel skyrmions in FGT due to a DMI tied to broken inversion symmetry from Fe vacancies.Additionally, the authors demonstrate the ability to drive the phase change optically, by locally heating the sample in-situ with a 520 nm pulsed laser during LTEM measurement.While the material parameters are marginally higher than previously reported in 2D vdW skyrmion systems, it seems to me that the novelty of this work comes from the mechanism/crystallography, which should be better described.The physics of the material is the new discovery here, as the material properties are almost the same as existing systems.I think this work needs to be more motivated by, and discuss, the physics of the system, rather than just experimental observations.
In, for example, (Fe0.5Co0.5)3GeTe2: • The presence of Neel skyrmions and THE of approximately the same value has previously been reported (Zhang et al.Room-temperature skyrmion lattice in a layered magnet (Fe0.5Co0.5)5GeTe2.Sci.Adv.8,eabm7103(2022).) • The ordering of the skyrmion lattice has previously been reported (Meisenheimer et al.Ordering of room-temperature magnetic skyrmions in a polar van der Waals magnet. Nat Commun 14, 3744 (2023).) • A similar phase diagram and thickness dependence has been reported (Zhang et al. Room-temperature skyrmion lattice in a layered magnet (Fe0.5Co0.5)5GeTe2.Sci. Adv.8,eabm7103(2022).) • Approximately the same Curie temperature and similar mechanism (ordering of empty Fe sites) has been reported (Zhang et al.A room temperature polar magnetic metal. Phys.Rev. Materials 6, 044403 (2022).) Author's reply: We sincerely appreciate the referee's comments regarding more indepth discussion on the mechanism/crystallography.These valuable comments are of great significance to improve the quality of our manuscript.In the last version of our manuscript, our primary focus is on reporting that the Feii vacancies in Fe3-xGaTe2 induces the displacement of Feii atoms, leading to the breaking of crystal inversion symmetry.These vacancies serve as the source of the Dzyaloshinskii-Moriya interaction, which not only contributes significantly to a room-temperature topological Hall effect but also facilitates the formation of small-sized Néel-type skyrmions with fs laser writability.Following the referee's comments, we have made a major revision to reinforce the discussion on the mechanism and crystallography.The key points of our accomplishments are outlined below, with more comprehensive discussions provided in subsequent responses to Referee B's Comments 2. We hope the referee will be satisfied with the revised manuscript as well as our responses.
(i) To understand the underlying physics for symmetry breaking on Feii sites, we have first conducted a thorough examination, employing improved ABF-STEM images and re-evaluating the single crystal XRD data.Our investigation has revealed that Fe deficiency is predominantly concentrated at the Feii positions.Additionally, a minor presence of asymmetric Fe deficiency at the Fei positions was observed, wherein the upper-layer Fei-a sites exhibit a higher degree of deficiency compared to the lower-layer Fei-b sites.
(ii) Upon establishing the crystal structure, we have conducted structure relaxation based on DFT calculations, considering Fei-a and Feii vacancies separately.Our computational results confirm that the asymmetric Fei-a vacancy primarily induces Feii deviation towards the c direction, which results in the symmetry breaking of the crystal, whereas the Feii vacancy exerts no influence on Feii deviation.In further response to the referee's comments, we'd like to show that there is indeed a functional difference between the T-B cycling and our in-site fs laser quenching approach.Typically, without a magnetic field, zero-field cooling can only result in the formation of interconnected, relatively long stripe domains, but does not spontaneously lead to the creation of skyrmions (as show in Fig. R5a-c) [Nature Communications 13.1 (2022): 3035].However, through our measurements with varying laser pulse fluences, we have identified the possibility of achieving skyrmion writing under zero magnetic field conditions.As depicted in Fig. R5a and R5d, under the condition of a single laser pulse fluence of 1.3 mJ/cm 2 , the stripe domains within the yellow box merely exhibit domain wall movement after the laser pulse.In Fig. R5b and R5e, when we increase the single laser pulse fluence to 9.4 mJ/cm 2 , the stripe domains become narrower and shorter, with some regions breaking to form skyrmions (highlighted in the red box).However, as we increase the single laser pulse fluence to 11 mJ/cm 2 , stripe domains are formed without skyrmions (Fig. R5c and R5f).These in-situ laser fluencedependent experiments indicate that a hybrid state with coexisting stripes and skyrmions is achievable without magnetic field.Currently, all the reported articles on fs laser-induced skyrmion writing have required external magnetic field assistance Author's reply: We sincerely thank the referee for careful reading of our manuscript.
These valuable suggestions and comments are greatly helpful for us to explore a more in-depth physical picture on the mechanism of the DMI.In response to these comments, we have supplemented additional structural characterizations and first-principles calculations, revealing that the asymmetric Fei vacancies are the primary cause of the  was used with electron-core interactions described by the projector augmented wave method for the pseudopotentials, and the exchange correlation energy calculated with the generalized gradient approximation of the Perdew-Burke-Ernzerhof (PBE) form.
The plane wave cutoff energy was 400 eV for all the calculations.In calculating the atomic shifts due to Fei-a and Feii vacancies, we used a 2×2 supercell and removed the bottommost Fei-a and Feii atoms.The Monckhorst-Pack scheme was used for the Γcentred 12 × 12 × 1 k-point sampling.All atoms' relaxations were performed until the force become smaller than 0.001 eV/A for determining the most stable geometries.

Response to "DFT is used to simulate the value of DMI, but is the relaxed structure comparable? Does this value match with what is measured (does it give
skyrmions/domains of similar size)?": The above ABF-STEM analysis has revealed the Fe deficiency is predominantly concentrated at the Feii positions, accompanied by a minor deficiency in asymmetric Fei-a positions.However, building such a vacancy model with exact occupancy ratio of Feii and Fei-a atoms requires a very large supercell structure, which is beyond the computation capability for DMI calculation.To reduce the computational complexity, the DFT calculation is based on the fixed crystal structure obtained from single-crystal XRD experiments, instead of a fully relaxed crystal structure optimized from DFT calculations.The reasons are described below: In considering the structural model in Fig. R8, the formation of DMI requires spatial inversion symmetry breaking in upper and lower triangles composed of Fei-Feii-Te atoms.In quantitative terms, the DMI vector can be expressed as where D is the DMI constant, uij represents the unit vector from Fei atom to Feii atom, and z represents the unit vector from magnetic Feii atom to heavy Te atom.It can be seen that if Feii is located at the center position of Te-Te atoms, where ever Fei-a or Feib are located, the upper D1 and lower D2 vectors would always cancel out with each other.This feature suggests that the spatial inversion symmetry breaking of Feii deviations is the primary cause of DMI, while Fei-a vacancies do not contribute DMI.
Thus, we can reasonably assume that atoms are fixed with full occupancy, and ignoring the steps of structure relaxation, which would otherwise necessitate the creation of impractically large supercells in the vacancy model.
In   relaxations with fixed δ(Feii) were performed with Gaussian smearing until the forces become smaller than 0.001 eV/Å.Next, spin-orbit coupling was included in the calculation, and the total energy of the system was determined as a function of the spin configuration as shown in Fig. R9a, and d|| equals to (EACW − ECW)/12.The DMI constant D was calculated using the equation D = 3√2d/(N F a 2 ) , where NF is the number of atomic layers, a is the lattice constant and d|| represents DMI strength.In the second step, the EDIFF is set to 10 -8 eV and the tetrahedron method with Blöchl corrections was used to get an accurate total-energy.The resulting relationship between D and δc(Feii − Ga) is presented in Fig. R9b.

Responses to: "What does the anisotropy of the parent compound look like?"
In order to compare the anisotropy of different Fe-content samples, we systematically have grown a series of Fe3-xGaTe2 single crystals by varying the Fe content in the raw material composition, utilizing a Te-flux method.To ensure the reliability of the compositions, EDS mapping was carried out at four distinct cleaved surfaces of these crystals.As surmised in Table R2, the chemical formulas for the crystals with the minimum and maximum Fe content correspond to Fe2.84±0.05GaTe2and Fe2.96±0.02GaTe2,respectively.This result indicates that Fe vacancies always exist in the single crystals synthesized using a Te-flux method.The chemical formula of   simulations.The initial zero-field skyrmion state was relaxed from a random state with 100 mT magnetic field.Employing the initial skyrmion state as the input, we systematically applied an in-plane magnetic field along y axis, and relaxed the magnetization to a stable state.

Referee B's Comment 4:
Why is there such a large variance in the sizes of the skyrmions?this is also different to what is generally reported.In fact, it almost looks bimodal in many images.
Author's reply: We thank the referee's comments on skyrmion size variance.In the LTEM experiments, we initially raised the sample temperature above Curie temperature, then cooled it to the target temperature with a 30 mT external magnetic field, and finally removed the external magnetic field to obtain zero-field skyrmions.As shown in Fig. R13a, the zero-field skyrmion density at low temperature (100 K) was relatively low, exhibiting non-uniformed size distribution.However, at higher temperature 250 K and 320 K, the zero-field skyrmion density gradually increased, and the size distribution became more uniform.
To clarify physical mechanism underlying the variation in skyrmion size at different temperatures, we simulated the zero-field skyrmions after field cooling (see Fig. R13 Note).As is known, the formation of skyrmions is determined by a delicate interplay of the magnetic parameters, including magnetic anisotropy Ku, DMI constant D, saturation magnetization Ms, sample thickness t, and exchange stiffness A. However, our experiments have demonstrated that increasing the sample temperature of Fe3-xGaTe2 leads to a significant reduction of magnetic anisotropy Ku, while other parameters keep nearly unchanged.Therefore, as shown in Fig. R13b, we decreased magnetic anisotropy constant Ku = 3.2  10 5 J/m 3 , 1.6  10 5 J/m 3 and 0.7  10 5 J/m 3 in the simulations to represent the increasing of sample temperature.Our simulations demonstrate that for large Ku at low temperature, the density of zero-field skyrmions is low, and the distant between the nearest skyrmions can be considerably large in certain regions, thus facilitating the expansion of skyrmion size upon the removal of the magnetic field.In contrast, with small Ku at high temperature, the skyrmions exhibit a densely hexagonal arrangement to each other, which suppresses the extension of the skyrmions.Consequently, they remain uniformly distributed after removing the magnetic field.
In summary, we discover that a larger Ku at low temperature results in lower skyrmion density, which exhibit a larger skyrmion-skyrmion distance and allow the expansion of skyrmion size.Consequently, the skyrmion size distribution at lower temperatures is non-uniform.In contrast, at higher temperatures, the lower Ku increases the skyrmion density and reduces the skyrmion-skyrmion distance, leading to a much more uniform size distribution.In the revised manuscript, we have incorporated discussions on skyrmion size variations into main text (refer to Page 13, Lines 358-360) and Supplementary information (refer to Supplementary Note 4 and Fig. S18).

Fig. R13 note:
To validate this experimental result, we conducted micromagnetic simulations of the field cooling process and then removing the external magnetic field.
Default magnetic parameters used in the simulations include A = 1.3 pJ/m, Ku = 0.8  10 5 J/m 3 , Ms = 2.5  10 5 A/m, D = 0.25 mJ/m 2 , and slab geometries with dimensions of 512  512  64, with a mesh size of 2  2  2 nm.Periodic boundary conditions were taken into account for large-scale simulations.The zero-field skyrmion state was relaxed from a random state with 30 mT magnetic field, and removing the magnetic field to relax until stable.

Referee B's Comment 5:
There needs to be more interpretation of the results and tying back to a structure-property relation for me to be comfortable recommending this paper.

Referee C's General Comment:
The authors investigate a non-stoichiometric room temperature magnet Fe2.86GaTe2 crystal where the Fe vacancies induce the formation of DMI by spatial inversion symmetry breaking.Such an in-plane isotropic DMI brings about RT Néel-type skyrmions, and the size of the skyrmions can be regulated by the sample thickness and the external magnetic field.The dynamic writing process of RT skyrmions in Fe2.86GaTe2 flakes enhances the potential application of spintronic devices.
The paper is timely and of interest.
Author's reply: We sincerely thank the referee for careful reading of our manuscript and pointing out that our paper is "timely and of interest".The further valuable suggestions and comments provided by the referee are greatly helpful to improve our manuscript.Below we answer the comments in a point-by-point basis.We hope the referee will be satisfied with the revised manuscript as well as our responses.

Referee C's Comment 1:
The Methods section describes the process for obtaining a Author's reply: We sincerely thank the referee for the valuable comments.In order to control Fe content, we have systematically grown a series of Fe3-xGaTe2 single crystals by varying the Fe content in the raw material composition, utilizing a Te-flux method.
Subsequently, comprehensive energy-dispersive X-ray spectroscopy (EDS) mapping was conducted on the cleaved surfaces of these crystals to determine their chemical composition.To ensure the reliability of the EDS results, mapping was carried out at four distinct areas for each sample.A comprehensive overview of the raw material composition and final product is outlined in Table R2.
Our experimental results clearly demonstrate that controlling the Fe content in the final crystals is achievable by varying the raw Fe ratio.Specifically, when the raw Fe ratio fall below 0.8, the Fe3-xGaTe2 phase cannot be formed.Instead, a mixture of phases, including GaTe and Ga2Te3 phases, is produced.In contrast, when the raw Fe ratio is equal to or greater than 0.9, Fe3-xGaTe2 single crystals can be crystallized, with the Fe content in these single crystals increasing proportionally with the raw Fe ratio.However, materials and the corresponding final crystal composition are listed in Table S1, Supplementary Fig. S1 and Fig. 1b.We found that the Fe deficiencies always exist in these crystals, while the minimum and maximum Fe contents correspond to Fe2.84±0.05GaTe2and Fe2.96±0.02GaTe2,respectively.This result implies the feasibility of inducing Fe deficiency in the samples.To highlight the existence of Fe vacancies, the subsequent studies were focused on the minimum Fe content sample Fe2.84±0.05GaTe2." Table R3.Summary of the raw material composition and the final product for the growth of Fe3-xGaTe2 samples using the self-flux method.Author's reply: We sincerely thank the referee for the valuable comments.In the previous version of our manuscript, we reported the observation of Néel-type skyrmions in the Fe3-xGaTe2 single crystals synthesized with a raw Fe ratio of 0.9.Their average chemical formula was denoted as Fe2.86GaTe2 for EDS mapping while Fe2.79GaTe2 for XRD refinement.It is crucial to highlight that all the crystals were grown using the same temperature conditions and raw material composition.The observed variation in the average chemical formula can be attributed to measurement errors associated with different characterization techniques.
In the revised manuscript, to enhance the accuracy of the chemical formula, an error bar has been added by summarizing the EDS results obtained at four different areas, and the chemical formula is denoted as Fe2.84±0.05GaTe2,which remains within the error range of the XRD refined chemical formula Fe2.79GaTe2.We have integrated error analysis of EDS determined chemical formula into both the main text (refer to Page 5, Lines 118-119) and Supplementary information (refer to Supplementary Note 1 and Table S1).
(see Fig. S19 and Supplementary Note 5 for the determination of magnetic parameters), which contribute to the reduction in skyrmion size (see Table S3, Fig. S20 and Supplementary Note 6 for the corresponding micromagnetic simulations)." Table R4.Magnetic parameters for skyrmion-host 2D materials.Author's reply: We sincerely thank the referee for the valuable comments.We have revised such expressions as recommended.Author's reply: We sincerely thank the reviewer for the valuable suggestions.We have replaced the figures as recommended.Referee B's Comment 2: I think there can be more explicit attention in the text to how pulsed writing is different from the quasi-thermodynamic state.E.g., as in the response: "Typically, without a magnetic field, zero-field cooling can only result in the formation of interconnected, relatively long stripe domains, but does not spontaneously lead to the creation of skyrmions (as show in Fig. R5a-c Author's reply: We sincerely thank the reviewer for the valuable suggestions.We have added the discussions as recommended.

Response to the Report of Referee
"Furthermore, conventional zero-field cooling can only result in the formation of interconnected, relatively long stripe domains, but does not spontaneously lead to the creation of skyrmions.To demonstrate the differences with in-site fs laser quenching approach, we conducted fluence-dependent laser pulse excitation without magnetic field."Referee B's Comment 3: The authors could also make the statement "spin clusters that contain topological defects such as skyrmionic and anti-skyrmionic nucleation centers (snapshot at t1) due to the ultrafast cooling at a quenching rate of up to 1012 K/s" (L457) more obvious to make the work read better.
Author's reply: We sincerely thank the reviewer for the valuable suggestions.We have revised the statements as recommended.
"As shown in Fig. 5d (see details in Movie S1), following the excitation by the femtosecond (fs) laser pulse, the initial melted spin state (snapshot at t0) rapidly evolves into numerous nanoscale spin clusters.These clusters contain topological defects, including skyrmionic and anti-skyrmionic nucleation centers (snapshot at t1).This transformation occurs due to the ultrafast cooling, achieved at a quenching rate of up to 10 12 K/s." Overall, the authors have rigorously answered the concerns in my initial review and I recommend the manuscript for publication.
Author's reply: We sincerely thank the reviewer for recommending our manuscript to be published in Nature Communications.

Referee C's General Comment:
In the revised manuscript and supplementary information, the authors have thoroughly characterised their sample, including EDS mapping and XRD data.In addition, they have made new discoveries regarding the Fei and Feii vacancies and provide a reasonable explanation for the symmetry breaking of the Fe2.84±0.05GaTe2crystal structure, which is also consistent with first-principles calculations.Overall, the authors have adequately addressed the reviewers' comments.
I recommend the manuscript for publication in Nature Communication.
Author's reply: We sincerely thank the reviewer for recommending our manuscript to be published in Nature Communications.The valuable suggestions and comments furnished by the referee are greatly helpful to improve our manuscript.
Fig. R1. a Experimental CBED patterns of Fe2.84±0.05GaTe2from [112 ̅ 0] direction.b Ray diagrams showing how increasing the C2 aperture size causes the CBED pattern to change from one in which individual disks are resolved to one in which all the disks overlap.

Referee A's Comment 3 :
The Fe columns in Fig.1fare not really clear.The atomic columns do not look clean.A better clear HAADF-STEM image will be better that shows better Fe Columns.Would the author claim that the Fe deficiency occurs only at those FeII positions?Author's reply: We highly appreciate the referee's comments.To provide a clearer view of the Fei and Feii columns, we have further acquired improved HAADF-and ABF-STEM images of the Fe2.84±0.05GaTe2sample along the [112 ̅ 0] zone axis, as shown in Fig.R3.To highlight the detailed information about the Feii columns, a magnified image was derived from the enclosed region of the ABF-STEM image, as shown in Fig.

Fig
Fig. R4. a Magnified ABF-STEM image of the single Fe2.84±0.05GaTe2layer.b Integrated imaging intensity line profile along the c-axis within the area marked by the Te-Feii-Te atoms in the blue rectangles.The red region indicates the Feii deviation from the centers of Te-Te atoms.c Integrated imaging intensity line profile of Fei-a-Fei-b atoms.Referee A's Comment 4: Can the author control the Fe content?

(
iii) Furthermore, we explored the correlation between Feii deviation values δc (Feii − Ga) and the DMI constant D based on the DFT calculations.It is found that no DMI is observed (the value of D is equal to zero) in a centrosymmetric structure without Feii deviation (δc = 0).However, once the Feii atom deviate from Ga plane (δc < 0) due to the asymmetric Fei-a vacancy, the inversion symmetry is broken and the value of D increases accordingly as the increase of the Feii deviation.It should be noted that the calculated value of D is comparable to that established in experiments, confirming the reliability of our structural model.We appreciate the referee's recommendation of the excellent work on (Fe0.5Co0.5)5GeTe2and polar van der Waals magnets.All relevant references have been appropriately cited in the main text to acknowledge the valuable contributions of the prior work in the field.Referee B's Comment 1: And while the in-situ measurement is new, its value would come from actual dynamical measurements-by doing just quasistatic quenching, it doesn't seem like there is functionally any difference between just T, B cycling without the optical pump (Meisenheimer et al.Ordering of room-temperature magnetic skyrmions in a polar van der Waals magnet.Nat Commun 14, 3744 (2023); Zhang et al.Room-temperature skyrmion lattice in a layered magnet (Fe0.5Co0.5)5GeTe2.Sci.Adv.8,eabm7103(2022)).Especially so because you motivate the experiment from the perspective of "ultrafast writing of skyrmions" (L 104, L 313).Author's reply: We are grateful for the referee's positive feedback and suggestions regarding the in-situ measurements.Femtosecond laser (fs) control of topological magnetic structures is a promising and still relatively unexplored field, involving complex physical processes such as ultrafast demagnetization [Nature Communications 14.1 (2023): 1378], optically induced magnetism [Physical Review Letters 125.26 (2020): 267205], optical-pumped spin dynamics, all-optical magnetization reversal, and more.It is widely known that, magnetic skyrmions in twodimensional materials are often metastable and typically require temperature-magnetic field (T-B) cycling [Nature Communications 13.1 (2022): 3035; Nano letters 22.19 (2022): 7804-7810], which is time-consuming and energy-intensive.In contrast, fs laser pulse-induced skyrmion writing, based on its unique quenching effect, offers the advantages of fast speed and low energy consumption [Nature Materials 20.1 (2021): 30-37].Moreover, it also allows for the adjustment of laser spot size and location, enabling selective writing in specific regions [Nano Letters 18.11 (2018): 7362-7371].Consequently, there has been significant interest in laser-induced change and switching of topological spin textures in recent years, which allows the exploration of metastability and hidden phases of topological spin textures [Science Advances 4.7 (2018): eaat3077].

[
Fig. R5.a, b, c.Ground states of the stripe domains obtained through zero-field cooling.d,e, f.Magnetic domain states after a single fs laser pulse with the fluence of 1.3 mJ/cm 2 , 9.4 mJ/cm 2 , and 11 mJ/cm 2 , respectively.The red boxes indicated isolated skyrmions after a single fs laser pulse.

Feii
Fig.R7aand R7b illustrate the scenario with no Fe vacancy.The electron density (colored in yellow) strongly overlaps between Feii-Ga atoms, forming the Feii-Ga honeycomb lattice plane with robust chemical bonding (highlighted by black dashed lines).Simultaneously, the Fei-a and Fei-b dimers are located at the center of the Feii-Ga honeycomb lattice but do not bond with Feii-Ga atoms.Consequently, the chemical bonding is mirror-symmetric along the Feii-Ga plane, leading to the absence of Feii displacements.Thus, perfect Fe3GaTe2 exhibits a centrosymmetric crystal structure with c → c mirror symmetry.

Fig. R7 .
Fig. R7.The relaxed crystal structure and corresponding electron density of Fe3-xGa atoms sliced from Fe3-xGaTe2 with a, b no vacancy, c, d Feii vacancy and e, f Fei-a vacancy.The black dashed line indicates the chemical bonding with electron-density overlapping.The yellow-colored electron densities are shown at the same isosurface value.
Fig. R8.Schematic illustration of DMI in asymmetric layers viewed from [112 ̅ 0] zone axis.The red arrow D1 represents the direction of DMI vector in the upper triangle composed of Fei-Feii-Te, while the blue arrow D2 represents the lower part in the opposite direction.The black arrow Deff represents the sum of the non-zero DMI vector.

Fig
Fig. R9.a, b Spin configurations implemented to calculate the DMI for clockwise (CW) and anticlockwise (ACW).c The calculated and experimental results for the relationship between the DMI and the Feii deviation.

Fig.
Fig. R9 note: The calculation of the DMI vector involved two steps.First, structural

Fig
Fig. R12 Note: Micromagnetic simulations were carried out using the GPU-accelerated micromagnetic simulation program Mumax 3 .Default magnetic parameters used in the simulations include A = 1.3 pJ/m, Ku = 0.8  10 5 J/m 3 , Ms = 2.5  10 5 A/m, D = 0.25 mJ/m 2 , and slab geometries with dimensions of 1024  1024  16, with a mesh size of 2  2  4 nm.Periodic boundary conditions were taken into account for large-scale

Fig
Fig. R13. a Lorentz Phase images of zero-field skyrmion after 30 mT field cooling at 100 K, 250 K and 320 K. b Micromagnetic simulations of zero-field skyrmion after 30

Fe2.86GaTe2 single
crystal, which was achieved directly by precisely controlling the initial molar ratio of the powder mixtures and growing conditions.I am inquiring about the method of determining the optimum molar ratio in this work?by experiment or theoretical calculation.And what is the advantage of using iron deficiency as a means of breaking the centrosymmetric structure compared to other methods, such as elemental doping (Ref.31)?
Fig. R14.Variation of simulated magnetic structure and corresponding skyrmion sizes with varied a, b DMI constant D, c, d saturation magnetization Ms, e, f sample thickness t, g, h magnetic anisotropy constant Ku, and i, j exchange stiffness A.

Fig. R14 note :
Fig. R14 note: Micromagnetic simulations were carried out using the GPU-accelerated micromagnetic simulation program Mumax 3 .Unless specified otherwise, default magnetic parameters used in the simulations include A = 4.0 pJ/m, Ku = 2.4  10 5 J/m 3 , Ms = 3.0  10 5 A/m, D = 0.90 mJ/m 2 , and slab geometries with dimensions of 512  512  64, with a mesh size of 2  2  2 nm.Periodic boundary conditions were taken into account for large-scale simulations.The initial skyrmion state was relaxed from a random state with 60 mT magnetic field.Employing the initial skyrmion state as the input, we systematically varied the magnetic parameters-D, Ms, t, Ku, and A individually-subsequently allowing the magnetization to evolve into a stabilized state through relaxation processes.
The authors have provided additional data to support the determination of the new phase.I am satisfied with the response, and I would recommend to proceed to publication.Just a note to the authors, it's easier to use CBED whole pattern and look into the HOLZ lines to check the symmetry.Author's reply: We sincerely thank the reviewer for recommending our manuscript to be published in Nature Communications.The valuable suggestions and comments furnished by the referee have significantly contributed to enhancing the quality of our manuscript.Response to theReport of Referee B Referee B's General Comment: The authors have clearly spent time answering my and the other reviewer's concerns and with the clarity that has been added to the introduction, namely the larger emphasis on the role of vacancy ordering, I am much more comfortable recommending this work for publication.Author's reply: We sincerely thank the reviewer for recommending our manuscript to be published in Nature Communications.Referee B's Comment 1: I think there are a few textual things that could be changed to increase the impact of the work, but the story and the physics are much clearer in the newer version.I think a subset of the SEAD patterns from Fig R2 should be added to Figure 1-the superlattice peaks are the strongest evidence for ordering and help explain the DMI.This could replace Fig 1g, since it is the same information but clearer.
The space group changed to non-inversion symmetry based on the XRD and HAADF-STEM image.It will be more convincing if additional data using CBED to confirm the space group.

Table R2 .
Summary of the raw material composition and the final product for the growth of Fe3-xGaTe2 samples using the self-flux method.